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<p>Classification of Types of Points:Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0,\tag{5.2.3}
\end{equation}
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<p class="continuation">A point <span class="process-math">\(x_0\)</span> such that <span class="process-math">\(P(x_0) \neq 0\)</span> is called an <dfn class="terminology">ordinary point</dfn>. If <span class="process-math">\(P(x_0)=0\text{,}\)</span> then <span class="process-math">\(x_0\)</span> is a <dfn class="terminology">singular point</dfn>. These are the definitions suitable when <span class="process-math">\(P(x), Q(x)\)</span> and <span class="process-math">\(R(x)\)</span> are polynomials and have no common factors.More generally, the ordinary and singular points are defined as:If at <span class="process-math">\(x=x_0\text{,}\)</span> both <span class="process-math">\(Q(x)/P(x)\)</span> and <span class="process-math">\(R(x)/P(x)\)</span> are analytic, then <span class="process-math">\(x=x_0\)</span> is called an <dfn class="terminology">ordinary point</dfn>. Otherwise, <span class="process-math">\(x=x_0\)</span> is a <dfn class="terminology">singular point</dfn>.</p>
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